Short Answer TypeA binary operation * on {1, 2} is a function from {1, 2} x {1, 2} to {1,2}, i.e., a function from {(1, 1), (1, 2), (2, 1), (2, 2)} → {1,2}.
Since 1 is the identity for the desired binary operation *,
* (1, 1) = 1, *(1, 2) = 2, * (2, 1) = 2 and the only choice left is for the pair (2, 2). Since 2 is the inverse of 2, i.e., * (2, 2) must be equal to 1. the number of desired binary operation is only one.
State whether the following statements are true or false. Justify.
(i) For an arbitrary binary operation * on a set N, a*a = a ∀ a ∈N.
(ii) If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a.
Consider a binary operation * on N defined as a * b = a3 + b3 . Choose the correct answer.
(A) Is * both associative and commutative ?
(B) Is * commutative but not associative?
(C) Is * associative but not commutative?
(D) Is * neither commutative nor associative?
Let A = Q x Q. Let be a binary operation on A defined by (a, b) * (c, d) = (ac, ad + b).
Then
(i) Find the identify element of (A, *)
(ii) Find the invertible elements of (A, *)
Define binary operation ‘*’ on Q as follows : a * b = a + b – ab, a, b∈ Q
(i) Find the identity element of (Q, *).
(ii) Which elements in (Q. *) are invertible?
is the inverse of a ≠ 0 for the multiplication operation ‘x’ on R.
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